Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics

Probability densities that are not uniquely determined by their moments are said to be “moment-indeterminate,” or “M-indeterminate.” Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.

     

The finite element approximation of a coupled reaction-diffusion problem with non-Lipschitz nonlinearities

Summary. A coupled semilinear elliptic problem modelling an irreversible, isothermal chemical reaction is introduced, and discretised using the usual piecewise linear Galerkin finite element approximation. An interesting feature of the problem is that a reaction order of less than one gives rise to a "dead core" region. Initially, one reactant is assumed to be acting as a catalyst and is kept constant. It is shown that error bounds previously obtained for a scheme involving numerical integration can be improved upon by considering a quadratic regularisation of the nonlinear term. This technique is then applied to the full coupled problem, and optimal and error bounds are proved in the absence of quadrature. For a scheme involving numerical integration, bounds similar to those obtained for the catalyst problem are shown to hold. Received May 25, 1993 / Revised version received July 5, 1994     

Extraction of ergosterol from peaty soils and determination by high performance liquid chromatography

The ergosterol content of soil can be used as an indicator of fungal activity. A method has been developed for the extraction and determination of ergosterol in organic soils, as part of a study to assess the correlation between fungal activity and the sequestration of metal pollutants. The moisture content of the soil affected the extraction process. Four consecutive extractions with methanol removed >95% of the ergosterol that can be obtained from the fresh sample (63% moisture) by exhaustive extraction. By freeze drying the soils prior to extraction (a) up to 35% more ergosterol was extracted after a single extraction, (b) >90% of the recoverable ergosterol was collected in two extractions and (c) the repeatability of the extraction was improved. Storage of soil extracts in the absence of light prevents degradation of ergosterol. A previously reported method for determination of ergosterol by HPLC has been improved by modification of the eluant composition. With 46% methanol/46% acetonitrile/8% dichloromethane, ergosterol was eluted with good resolution approximately 8 min after injection of 20 mul of the extract. The detection limit of the HPLC method was 0.5 mug/ml ergosterol, equivalent to 0.06 mug/g in 25 g fresh soil. Changes in ergosterol contents of peaty soil treated with fungicide, and in samples of the peaty podzol and a humus iron podzol in the vicinity of fungal fruiting bodies, have been determined.     

Development of Molecular Mechanics Torsion Parameters for alpha,beta-Cyclopropyl Ketones and Conformational Analysis of Bicyclo[m.1.0]alkan-2-ones

Conformations of cyclopropyl methyl ketone have been studied using ab initio methods in an effort to quantify the effects of conjugative overlap between the cyclopropane ring and an adjacent ketone carbonyl. Results were comparable with previous experimental and theoretical studies. Cyclopropyl methyl ketone exhibits a global energy minimum in the s-cis conformer and a local energy minimum near the s-trans conformer. The potential energy curve obtained was used to derive torsion parameters which were employed in molecular mechanics studies of the conformations of the set of bicyclo[m.1.0]alkan-2-ones having larger ring sizes from five- to 16-membered. Similar conformations for the cyclopropyl ketone substructure are observed for all the medium and large ring systems examined. Possible synthetic ramifications of local conformational anchoring by this functional group array are discussed.     

Shape-tunable polymer nodules grown from liposomes via ring-opening metathesis polymerization

A new inisurf (acting as surfactant and initiator) molecule for ring-opening metathesis polymerization (ROMP) was synthesized and used in aqueous solution in order to control the size and shape of polymer nodules grown from liposomes. Nodules were observed to grow in size with conversion of monomer, and depending on the monomer used, they adopted either a spherical or comet-like shape. Here, we investigate polymer production from a liposome surface. We use a hydrophobic derivative of the Grubbs catalyst positioned at the liposome surface to allow for ROMP of monomers dissolved in the aqueous outer phase. We obtain nodules of polymer that can grow up to tens of micrometers, unveiling new efficient possibilities of polymerization from a membrane in an aqueous solution.     

N2 activation by iron-sulfur complexes

Inspired by the determination of the structure of the nitrogenase enzyme cofactor by Rees et al., the binding of an N₂ molecule to some model iron-sulfur compounds was investigated usingab initio calculations. Side-on and end-on coordination to one two and four iron centers were investigated. In most cases, the N₂ is loosely bound and retains its internal triple bond, but a few examples are found where the N₂ is “activated” and has a longer N-N bond length.     

Matrix-assisted laser desorption/ionisation mass spectrometry for the direct analysis of enzymatically digested kappa- iota- and hybrid iota/nu-carrageenans

Enzymatically digested oligosaccharides of kappa-, iota- and hybrid iota/nu-carrageenans were analysed using matrix-assisted laser desorption/ionisation time-of-flight mass spectrometry in the negative-ion mode. nor-Harmane was used as matrix. Depending on the stock concentration and the laser intensity applied, the oligosaccharides exhibited losses of sulphate units (neutralised by the Na+ ion, and thus non-stable), leaving the primary backbone structure in most cases with only the deprotonated sulphate groups (carrying the negative charge, stable). This meant that kappa- and iota-oligosaccharides could not be easily distinguished from one another since they share the same primary backbone structure. However, for the hybrid iota/nu-oligosaccharides the primary backbone structure could be identified since the nu-carrageenan repeating unit differs from that of the kappa/iota-carrageenan unit. For all types of oligosaccharides, the results indicated cleavage of an anhydrogalactose unit from the non-reducing end. Specifically, for the hybrid oligosaccharides of iota/nu-carrageenans, this type of fragmentation means that the nu-carrageenan unit is not positioned on the non-reducing end of the hybrid oligosaccharides. Dehydration reactions, and exchange reactions of Na+ with K+ and Ca2+, were also observed.     

Interdependence of redox state,hydrogen bonding,anion recognition and charge partition in crystals of (EDT-TTF-CONHMe)6 [Re6Se8(CN)6] (CH3CN)2(CH2Cl2)2

Neutral pi-conjugated molecules and their radical cations co-exist in [(EDT-TTF-CONHMe+*)4(EDT-TTF-CONHMe0)2] [Re6Se8(CN)6]4- (CH3CN)2(CH2Cl2)2 whose crystal structure reveals that, upon one-electron oxidation, an activation of the N-H and C-H hydrogen bond donor ability is coupled to a deactivation of the hydrogen bond acceptor character of the carbonyl oxygen atom: this is expressed in the supramolecular hydrogen bond pattern and, ultimately, into charge localisation and partition in the solid state.     

Mechanism of the aromatic hydroxylation of thiophene by acid-catalyzed peracid oxidation

The oxidation of thiophene (1) with peracids in a strongly acidic environment yielded thiophen-2-one (4) as the product of an apparent direct hydroxylation of the thiophene aromatic ring together with the anticipated thiophene-S-oxide dimers, 2a,b, as the main products. Formation of the latter dimers can be rationalized in a straightforward manner by initial oxidation at the sulfur atom of thiophene (1) to yield thiophene-S-oxide followed by subsequent dimerization in a Diels-Alder type reaction. Trapping experiments in the presence of a competing dienophile indicated that thiophen-2-one (4) did not originate from the monomeric thiophene-S-oxide but was the product of an independent reaction pathway. The extent of thiophen-2-one (4) formation correlated with the acidity of the reaction medium and was suppressed in the presence of water, the latter presumably acting as a competing base. As evidenced by the use of 2,5-dideuterated thiophene (1-D), its mechanism of formation involved a 1,2-hydride shift, a feature commonly described in the peracid-mediated epoxidation of aromatic hydrocarbons and indicative for the occurrence of cationic intermediates. In agreement with all these observations we propose a mechanism involving initial protonation of thiophene followed by nucleophilic attack of the peracid in position 2 of the thiophene ring. Intramolecular epoxidation may lead to the formation of thiophene 2,3-epoxide as a highly reactive intermediate that then undergoes heterolytic ring opening and a 1,2-hydride shift to yield thiophen-2-one (4) after a final, acid-catalyzed, isomerization of the double bond.